Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-x-6y &= -1 \\ -2x-9y &= -6\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-2x = 9y-6$ Divide both sides by $-2$ to isolate $x$ $x = {-\dfrac{9}{2}y + 3}$ Substitute this expression for $x$ in the first equation. $-({-\dfrac{9}{2}y + 3}) - 6y = -1$ $\dfrac{9}{2}y - 3 - 6y = -1$ Simplify by combining terms, then solve for $y$ $-\dfrac{3}{2}y - 3 = -1$ $-\dfrac{3}{2}y = 2$ $y = -\dfrac{4}{3}$ Substitute $-\dfrac{4}{3}$ for $y$ in the top equation. $-x-6( -\dfrac{4}{3}) = -1$ $-x+8 = -1$ $-x = -9$ $x = 9$ The solution is $\enspace x = 9, \enspace y = -\dfrac{4}{3}$.